Coherent Density Fluctuation Model Research Articles

Temperature Dependence of the Symmetry Energy Components for Finite Nuclei

We investigate the temperature dependence of the volume and surface components of the nuclear symmetry energy (NSE) and their ratio in the framework of the local density approximation. The results of these quantities for finite nuclei are obtained within the coherent density fluctuation model (CDFM). The CDFM weight function is obtained using the temperature-dependent proton and neutron densities calculated through the HFBTHO code that solves the nuclear Skyrme-Hartree-Fock-Bogoliubov problem by using the cylindrical transformed deformed harmonic-oscillator basis. We present and discuss the values of the volume and surface contributions to the NSE and their ratio obtained for the Ni, Sn, and Pb isotopic chains around double-magic 78Ni, 132Sn, and 208Pb nuclei. The results for the T-dependence of the considered quantities are compared with estimations made previously for zero temperature showing the behavior of the NSE components and their ratio, as well as with the available experimental data. The sensitivity of the results on various forms of the density dependence of the symmetry energy is studied. We confirm the existence of “kinks” of these quantities as functions of the mass number at T = 0 MeV for the double closed-shell nuclei 78Ni and 132Sn and the lack of “kinks” for the Pb isotopes, as well as the disappearance of these kinks as the temperature increases.

Effect of temperature on the volume and surface contributions in the symmetry energy of rare earth nuclei

The effect of temperature on the volume and surface contributions in the nuclear symmetry energy and their ratio in the isotopic chains of rare earth Nd, Sm, Gd, and Dy nuclei with N = 82-126 is analyzed in the framework of coherent density fluctuation model (CDFM). The weight function of nuclei, within CDFM, are calculated by using the densities from the temperature-dependent relativistic mean-field (RMF) model. Firstly, we discuss the temperature-dependence of bulk properties of nuclei, within RMF model, such as binding energy, deformation parameter, charge radius and isotopic shift along with comparison with the available experimental data at temperature T=0 MeV. Further, we discuss the thermal evolution of symmetry energy and its volume and surface components. At T=0 MeV, the persistence of a peak in the symmetry energy and components at neutron number N=100 shows the manifestation of deformed shell closure in consonance with an earlier study by one of us [L. Satpathy, S.K. Patra, J. Phys. G 30 (2004) 771; S.K. Ghorui, et al., Phys. Rev. C 85 (2012) 064327]. However, the scenario changes with rise in temperature and the magnitude of peak decreases at higher temperatures. At T=3 MeV, the peak disappears which may be due to shape change in addition to quenching of shell effects since the quadrupole deformation parameter β2 decreases with an increase in temperature and nuclei become spherical at T=3 MeV. It indicates that behavior of symmetry energy is closely related to the deformation/shape of the nuclei. We have also discussed the values of volume symmetry energy, surface symmetry energy and their ratio which are in consonance with available experimental data.

ALPHA PARTICLE MOMENTUM DISTRIBUTION IN NUCLEI WITHIN THE COHERENT DENSITY FLUCTUATION MODEL

The alpha particle centre of mass momentum distribution in nuclei is determined on the basis of the four-body density matrix obtained within the coherent density fluctuation model. The calculations are carried out for a number of nuclei. The results are compared with those deduced from analyses of experimental data on alpha-particle knockout reactions induced by electrons,protons and alpha-particles which provide information on the alpha-particle momentum distribution.

Effective surface properties of light, heavy, and superheavy nuclei

Starting from light to superheavy nuclei, we have calculated the effective surface properties such as the symmetry energy, neutron pressure, and symmetry energy curvature using the coherent density fluctuation model. The isotopic chains of O, Ca, Ni, Zr, Sn, Pb, and Z = 120 are considered in the present analysis, which cover nuclei over the whole nuclear chart. The matter density distributions of these nuclei along with the ground state bulk properties are calculated within the spherically symmetric effective field theory motivated relativistic mean field model by using the recently developed IOPB-I, FSUGarnet, and G3 parameter sets. The calculated results are compared with the predictions of the widely used NL3 parameter set and found to be in good agreement. We observe a few signatures of shell and/or sub-shell structure in the isotopic chains of nuclei. The present investigations are quite relevant for the synthesis of exotic nuclei with high isospin asymmetry, including superheavy nuclei, and also to constrain an equation of state of nuclear matter.

Density dependence of symmetry energy in deformed 162Sm nucleus

The density dependence of symmetry energy in deformed [Formula: see text]Sm nucleus is discussed in the context of conversion of deformed density into spherical equivalent one by using the different methods. The results present a variation in the symmetry energy, using the spherical equivalent density extracted by different procedures, within the coherent density fluctuation model.

Correlation among the nuclear structure and effective symmetry energy of finite nuclei

The structural evolution and correlation of ground state properties with the effective nuclear matter characteristics for the even-even isotopes of Th and U nuclei are investigated within axially and triaxially deformed Relativistic-Hartree-Bogoliubov approaches. The density dependent meson exchange (DD-ME), point coupling (DD-PC) and non-linear (NL3⁎) force parameters are used for the present analysis. The potential energy surface is calculated by the quadratic constraint method to analyze the structural evolution by mean of shape transitions over an isotopic chain. The Coherent Density Fluctuation Model is used to formulate the nuclear matter characteristics such as symmetry energy and neutron pressure by weightening the density for finite nuclei. Further, we have predicted the possible modes of decay (α-decay and/or β-decay) in order to understand the stability of these nuclei. The calculated results are compared with experimental and other theoretical models, wherever available. We found a peak for effective symmetry energy and neutron pressure at N=150 for all the force parameters, which may correlate with the shell/sub-shell closures in an isotopic chain.

Proton momentum distributions and elastic electron scattering form factors for some Ge isotopes

The proton momentum distributions (PMD) and the elasticelectron scattering form factors F(q) of the ground state for someeven mass nuclei in the 2p-1f shell for 70Ge, 72Ge, 74Ge and 76Ge arecalculated by using the Coherent Density Fluctuation Model (CDFM)and expressed in terms of the fluctuation function (weight function)|F(x)|2. The fluctuation function has been related to the chargedensity distribution (CDD) of the nuclei and determined from thetheory and experiment. The property of the long-tail behavior at highmomentum region of the proton momentum distribution has beenobtained by both the theoretical and experimental fluctuationfunctions. The calculated form factors F (q) of all nuclei under studyare in good agreement with those of experimental data throughout allvalues of momentum transfer q.

Infinite nuclear matter characteristics of the finite nuclei within relativistic mean‐field formalism

We investigate the mass dependence of the nuclear structure and the effective symmetry energy for the Kr and Ba nuclei. The axially deformed self‐consistent relativistic mean field with the nonlinear NL3* parameter set is used in the analysis. The coherent density fluctuation model is adopted to formulate the nuclear matter observables such as symmetry energy, neutron pressure, and symmetry energy curvature of finite nuclei as a function of the radius. We found a notable signature of the shell/subshell closure at N = 50 and 82 for the isotopic chains of Kr and Ba nuclei, respectively.

Temperature dependence of the volume and surface contributions to the nuclear symmetry energy within the coherent density fluctuation model

The temperature dependence of the volume and surface components of the nuclear symmetry energy (NSE) and their ratio is investigated in the framework of the local density approximation (LDA). The results of these quantities for finite nuclei are obtained within the coherent density fluctuation model (CDFM). The CDFM weight function is obtained using the temperature-dependent proton and neutron densities calculated through the HFBTHO code that solves the nuclear Skyrme-Hartree-Fock-Bogoliubov problem by using the cylindrical transformed deformed harmonic-oscillator basis. We present and discuss the values of the volume and surface contributions to the NSE and their ratio obtained for the Ni, Sn, and Pb isotopic chains around double-magic $^{78}$Ni, $^{132}$Sn, and $^{208}$Pb nuclei. The results for the $T$-dependence of the considered quantities are compared with estimations made previously for zero temperature showing the behavior of the NSE components and their ratio, as well as with the available experimental data. The sensitivity of the results on various forms of the density dependence of the symmetry energy is studied. We confirm the existence of `kinks' of these quantities as functions of the mass number at $T=0$ MeV for the double closed-shell nuclei $^{78}$Ni and $^{132}$Sn and the lack of `kinks' for the Pb isotopes, as well as the disappearance of these kinks as the temperature increases.

Symmetry Energy and Its Components in Finite Nuclei

We derive the volume and surface components of the nuclear symmetry energy (NSE) and their ratio within the coherent density fluctuation model. The estimations use the results of the model for the NSE in finite nuclei based on the Brueckner and Skyrme energy-density functionals for nuclear matter. The obtained values of the volume and surface contributions to the NSE and their ratio for the Ni, Sn, and Pb isotopic chains are compared with estimations of other approaches which have used available experimental data on binding energies, neutron-skin thicknesses, and excitation energies to isobaric analog states (IAS). Apart from the density dependence investigated in our previous works, we study also the temperature dependence of the symmetry energy in finite nuclei in the framework of the local density approximation combining it with the self-consistent Skyrme-HFB method using the cylindrical transformed deformed harmonic-oscillator basis. The results for the thermal evolution of the NSE in the interval T = 0–4 MeV show that its values decrease with temperature. The investigations of the T-dependence of the neutron and proton root-mean-square radii and the corresponding neutron skin thickness point out that the effect of temperature leads mainly to a substantial increase of the neutron radii and skins, especially in nuclei which are more rich of neutrons.

Elastic Electron Scattering from <sup>88</sup>Sr and <sup>89</sup>Y Nuclei

The ground state proton momentum distributions and elastic charge form factors for ⁸⁸Sr and ⁸⁹Y nuclei have been derived and studied using the Coherent Density Fluctuation Model and formulated by means of the fluctuation function (weight function) |<i>f(x)</i>|² The fluctuation function has been connected to the charge density distribution of the nuclei and determined from the theory and experiment. The feature of the long-tail behavior at high momentum region of the PMD has been calculated by both the theoretical and experimental fluctuation functions. The calculated form factors F (q) of all nuclei under study are in good agreement with those of experimental data throughout all values of momentum transfer q.

Volume and surface contributions to the nuclear symmetry energy within the coherent density fluctuation model

The volume and surface components of the nuclear symmetry energy (NSE) and their ratio are calculated within the coherent density fluctuation model (CDFM). The estimations use the results of the model for the NSE in finite nuclei based on the Brueckner energy-density functional for nuclear matter. In addition, we present results for the NSE and its volume and surface contributions obtained by using the Skyrme energy-density functional. The CDFM weight function is obtained using the proton and neutron densities from the self-consistent HF+BCS method with Skyrme interactions. We present and discuss the values of the volume and surface contributions to the NSE and their ratio obtained for the Ni, Sn, and Pb isotopic chains studying their isotopic sensitivity. The results are compared with estimations of other approaches which have used available experimental data on binding energies, neutron-skin thicknesses, excitation energies to isobaric analog states (IAS) and also with results of other theoretical methods.

Structure of Exotic Mg Isotopes and Temperature Dependence of the Symmetry Energy of Finite Nuclei

We study various ground-state properties of neutron-rich and neutron-deficient Mg isotopes with A=20-36 in the framework of the self-consistent deformed Skyrme-Hartree- Fock plus BCS method. The nuclear symmetry energy is investigated for the same isotopic chain following the theoretical approach based on the coherent density fluctuation model. The results of the calculations show that the behavior of the nuclear charge radii and the nuclear matter properties in the Mg isotopic chain is closely related to the nuclear deformation. The temperature dependence of the symmetry energy for isotopic chains of even-even Ni (A=58- 82), Sn (A=124-152), and Pb (A=202-214) nuclei is investigated in the framework of the local density approximation. The results for the thermal evolution of the symmetry energy coefficient show that for all isotopic chains considered and for both Skyrme forces used in the calculations the symmetry energy coefficient decreases with the increase of the mass number in the temperature interval T = 0-5 MeV.

Nucleon Momentum Distributions and Elastic Electron Scattering from 59Co, 61Ni, 63Cu and 65Cu Nuclei

The nucleon momentum distributions (NMD) and the elastic electron scattering form factors F(q) of the ground state for some 1f-2p shell nuclei for 59Co, 61Ni, 63Cu and 65Cu are calculated using the Coherent Density Fluctuation Model (CDFM) and expressed in terms of the fluctuation function (weight function) . The fluctuation function has been related to the nucleon density distribution (NDD) of the nuclei and determined from the theory and experiment. The property of the long-tail behavior at high momentum region of the NMD has been obtained by both the theoretical and experimental fluctuation functions. The calculated form factors of all nuclei under study are in good agreement with those of experimental data throughout all values of momentum transfer

Nucleon momentum distributions and elastic electron scattering from 19F, 25Mg, 27Al, and 29Si nuclei

The nucleon momentum distributions and elastic electron scattering form factors of the ground state for some odd 2s–1d shell nuclei, such as 19F, 25Mg, 27Al, and 29Si, have been investigated using the coherent density fluctuation model and expressed in terms of the fluctuation function (weight function) |f(x)|2. The fluctuation function has been related to the nucleon density distribution of the nuclei and determined from the theory. The property of the long-tail manner at high-momentum region of the nucleon momentum distribution has been obtained by theoretical fluctuation function. The calculated form factors F(q) of all nuclei under study are in very good agreement with those of experimental data throughout all values of momentum transfer q. It is concluded that the contributions of the quadrupole form factor F C2(q) in 25Mg and 27Al nuclei, which are characterized by the undeformed 2s–1d shell model, are necessary for getting a remarkable agreement between the theoretical and experimental form factors.

Symmetry Energy and Structure of Exotic Nuclei

The symmetry energy, the neutron pressure and the asymmetric compressibility of spherical Ni, Sn, Pb and deformed Kr, Sm neutron-rich even-even nuclei are calculated within the coherent density fluctuation model using the symmetry energy as a function of density within the Brueckner energy-density functional. The correlation between the thickness of the neutron skin and the characteristics related with the density dependence of the nuclear symmetry energy is investigated for isotopic chains of these nuclei in the framework of the deformed self-consistent mean-field Skyrme HF+BCS method. The mass dependence of the nuclear symmetry energy and the neutron skin thickness are also studied together with the role of the neutron-proton asymmetry. The studied correlations reveal a smoother behavior in the case of spherical nuclei than for deformed ones. We also notice that the neutron skin thickness obtained for 208Pb with SLy4 force is found to be in a good agreement with the recent data. In addition to the interest that this study may have by itself, we give some numerical arguments in proof of the existence of peculiarities of the studied quantities in Ni and Sn isotopic chains that are not present in the Pb chain.

Ground-state properties and symmetry energy of neutron-rich and neutron-deficient Mg isotopes

A comprehensive study of various ground-state properties of neutron-rich and neutron-deficient Mg isotopes with $A$=20-36 is performed in the framework of the self-consistent deformed Skyrme-Hartree-Fock plus BCS method. The correlation between the skin thickness and the characteristics related with the density dependence of the nuclear symmetry energy is investigated for this isotopic chain following the theoretical approach based on the coherent density fluctuation model and using the Brueckner energy-density functional. The results of the calculations show that the behavior of the nuclear charge radii and the nuclear symmetry energy in the Mg isotopic chain is closely related to the nuclear deformation. We also study, within our theoretical scheme, the emergence of an "island of inversion" at neutron-rich $^{32}$Mg nucleus, that was recently proposed from the analyses of spectroscopic measurements of $^{32}$Mg low-lying energy spectrum and the charge rms radii of all magnesium isotopes in the $sd$ shell.

Structure properties of medium and heavy exotic nuclei

Investigations of important characteristics of the structure of nuclei near drip-lines in coordinate and momentum space have been performed. The charge form factors, charge and matter densities and the corresponding rms radii for even-even isotopes of Ni, Kr, and Sn are calculated in the framework of deformed self-consistent mean field Skyrme DDHF+BCS method. The resulting charge radii and neutron skin thicknesses of these nuclei are compared with available experimental data, as well as with other theoretical predictions. The formation of a neutron skin is analyzed in terms of various definitions. Its correlation with the nuclear symmetry energy is studied within the coherent density fluctuation model using the symmetry energy as a function of density within the Brueckner energy-density functional. The nucleon momentum distributions for the same isotopic chains of neutron-rich nuclei are studied in the framework of the same mean-field method, as well as of theoretical correlation methods based on light-front dynamics and local density approximation. The isotopic sensitivities of the calculated neutron and proton momentum distributions are investigated together with the effects of nucleon correlations and deformation of nuclei.

Symmetry energy of deformed neutron-rich nuclei

The symmetry energy, the neutron pressure, and the asymmetric compressibility of deformed neutron-rich even-even nuclei are calculated for the examples of Kr and Sm isotopes within the coherent density fluctuation model using the symmetry energy as a function of density within the Brueckner energy-density functional. The correlation between the thickness of the neutron skin and the characteristics related to the density dependence of the nuclear symmetry energy is investigated for isotopic chains of these nuclei in the framework of the self-consistent Skyrme-Hartree-Fock plus BCS method. Results for an extended chain of Pb isotopes are also presented. A remarkable difference is found in the trend followed by the different isotopic chains: the studied correlations reveal a smoother behavior in the Pb case than in the other cases. We also notice that the neutron skin thickness obtained for ${}^{208}$Pb with the SLy4 force is found to be in a good agreement with recent data.

Nucleon momentum distributions and elastic electron scattering form factors for some 1p-shell nuclei

The nucleon momentum distributions (NMD) and elastic electron scattering form factors of the ground state for 1p-shell nuclei with Z = N (such as 6Li, 10B, 12C and 14N nuclei) have been calculated in the framework of the coherent density fluctuation model (CDFM) and expressed in terms of the weight function \(\left| {f( x )} \right|^2\). The weight function has been expressed in terms of nucleon density distribution (NDD) of the nuclei and determined from the theory and the experiment. The feature of the long-tail behaviour at high-momentum region of the NMDs has been obtained by both the theoretical and experimental weight functions. The experimental form factors F(q) of all the considered nuclei are very well reproduced by the present calculations for all values of momentum transfer q. It is found that the contributions of the quadrupole form factors FC2(q) in 10B and 14N nuclei, which are described by the undeformed p-shell model, are essential for obtaining a remarkable agreement between the theoretical and experimental form factors.